Relative expanders or weakly relatively Ramanujan graphs. (Q1421136)

Let \(G\) be a fixed graph with largest eigenvalue \(\lambda_0\) and with universal cover having spectral radius \(\rho\). The author demonstrates that a random cover of large degree over \(G\) has its new eigenvalues bounded in absolute value by roughly \(\sqrt{\lambda_0\, \rho}\). The main result is a ``relative version'' of the Broder-Shamir bound on eigenvalues of random regular graphs.